Intuitive diagrams for models of non-well founded set theory Based on our intuition about von-Neuman's rank, there is a standard view to describe a model of ZFC as a large V-shape world. 
When we remove the Axiom of Foundation (AF) from ZFC and replace it with an anti-foundation axiom like (AFA, BAFA, SAFA, etc.) we expand the V-shape world of ZFC by adding some new non-well founded sets. I wonder what is the true intuition about these expanded worlds? How should we imagine them? Which one of these descriptions are more useful?

Question: Please introduce some references for intuitive diagrams which describe the shape of the world of theories ZFC-AF+AFA, ZFC-AF+BAFA & ZFC-AF+SAFA. Any reference for diagrams which describe the shape of other non-well founded axiomatic systems which are essentially different from ZFC like NF and NFU are also welcome.   

 A: Hope this reference helps:  "Broadening the Iterative Conception of Set"  by Mark F. Sharlow, Notre Dame Journal of Formal Logic, Volume 42, Number 3, 2001, pp.149-170.  According to the abstract, "the modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets.  It is suggested that this modified iterated conception of set supports the axioms of Quine's set theory NF".  Pay particular attention to Section 4--"Loosening Up the hierarchy" and figures 1-4.
As regards the relation of the iterative concept of set with, say, AFA, I refer you to a quote from Stephen G. Simpson regarding this very topic (this from one of his postings):
"AFA in no way invalidates the iterative concept of set.  The intended model V of ZFC (including the axiom of foundation) is a canonical inner submodel of the intended model $V^{*}$ of 
$ZFC^{*}$=ZFC-Foundation +Anti-foundation [AFA--my comment].
Namely, V is the well-founded part of $V^{*}$.  And all of the f.o.m. action takes place in V.  And each of V and $V^{*}$ is canonically recoverable from the other.  (The elements of $V^{*}$ are just the isomorphism types of directed graphs in V [this suggests a way (at least to me) to generalize the iterative concept of set--let each stage of the hierarchy form the isomorphism types of new subdigraphs, including the ones that violate the Axiom of Foundation--my comment].)  If we refer to the elements of V as 'sets' and the elements of $V^{*}$ as 'hypersets' (Sazonov) or 'schmets' (Anderson) or whatever, there is no conflict." (www.cs.nyu.edu/pipermail/fom/2000-May/004007.html)
At least it is something to think about.         
